An active noise cancellation system of the kind described above, having several sound sources transmitting noise cancellation signals and several sensors receiving residual noise, has been suggested by S. J. Elliott, I. A. Stothers and P. A. Nelson in their article "A Multiple Error LMS Algorithm and its Application to the Active Control of Sound and Vibration", IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-35, No. 10, October, 1987. A block diagram illustrating such a system is presented also in FIG. 1 of the accompanying drawing, whereto reference will be made in the following.
Let us assume that the system includes L loudspeakers 2 and M microphones 3. A reference signal x(n) is fed to L adaptive filters 1, each having a transfer function W(i,n), by which is meant the transfer function i at time n. In the following, the notation w(i,j,n) means the coefficient j of the transfer function i modelled with an FIR filter at time n. Let the length of these transfer functions be I. The outputs y(k,n) of these filters - - - this is thus the output signal of the transfer function W(k,n) - - - are fed to the L loudspeakers 2. Let the transfer function from loudspeaker i to microphone j be C(i,j,n), and thus its coefficient k at time n is c(i,j,k,n). Let us further model these transfer functions with FIR filters, and let the length of each of these filters be J. The signals from each of the loudspeakers 2 are received by each of the microphones 3. The signal from the m microphone 3 is e(m,n), which is the sum of signals from all loudspeakers 2, plus the unattenuated noise d(m,n). This situation is illustrated in the block diagram of FIG. 2 of the accompanying drawing, where, for clarity, only one of the microphones 3 is shown.
On the basis of the starting assertions presented above, the following equations can be derived: ##EQU1##
If the squares of the expected values of all microphone signals are defined as the total noise N.sub.tot in the space in which the noise cancellation signals are adapted to function, that is, in the target area, the following equation is obtained: ##EQU2##
The differential of the total error with respect to the coefficient i of the transfer function W(l,i) is ##EQU3##
Let us assume that W(l,n) and C(l,m,n) are for a moment time-invariant. This means in practice that they are changing only slowly compared with the reference signal x(n) and the residual noise d(m,n). Then the transfer function W(l,n) is denoted as W(l) and the transfer function C(l,m,n) as C(l,m). Correspondingly, the i:th coefficients of said functions are denoted as w(l,i) and c(l,m,i). Differentiating equation 3 gives ##EQU4##
Let us further assume that we have estimates of the transfer functions C(l,m) available, and let us denote these estimates as C'(l,m). If each coefficient w(l,i) of the transfer function W(l) is adjusted at every sample time by a quantity proportional to the negative instantaneous value of the differential given by formula 5, a modified multi-channel filtered-x type algorithm for the coefficient w(l,i,n+l) of the transfer function is obtained, which thus represents the value of said coefficient at a new time n+l. ##EQU5## where .alpha. is the adaptation coefficient.
The algorithm recounted above has been implemented in a real-time prototype and its performance measured. This has been reported in the above-stated article by Elliott et al. Substantial noise cancellation was only found at the frequency of the reference signal. An essential problem of the algorithm described above and the system based thereon is that fixed transfer functions are used for the estimation of the transmission paths between the loudspeakers and the sensors. In a multi-channel system this entails the need to measure several transfer functions for each installation. For instance, using four loudspeakers and eight sensors requires measurement of the transfer functions of 32 different transmission paths, which for practical reasons is not at all simple. In addition, the use of fixed transfer function estimates makes the system incapable of responding to changes in the acoustics of the target area, such as variations in the number and position of passengers if the target area is a vehicle, variations in temperature and humidity, or changes due to component ageing or failure.